Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect
For over a hundred years, the model of prey–predator has been handled extensively. It has been proved that Lotka–Volterra (LV) models are not stable from all previous studies. In these models, there is divergent extinction of one species or cyclic oscillation. There is no solution for asymptotic sta...
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| Format: | Article |
| Language: | English |
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Wiley
2023-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2023/8795660 |
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| author | A. El-Mesady Omar Bazighifan Serkan Araci |
| author_facet | A. El-Mesady Omar Bazighifan Serkan Araci |
| author_sort | A. El-Mesady |
| collection | DOAJ |
| description | For over a hundred years, the model of prey–predator has been handled extensively. It has been proved that Lotka–Volterra (LV) models are not stable from all previous studies. In these models, there is divergent extinction of one species or cyclic oscillation. There is no solution for asymptotic stability for the prey–predator models. In this paper, a Caputo fractional-order one prey-two predators’ model with an immigration effect is investigated. Then, the effect of adding a small immigration term to the prey or predators’ populations is studied. The extension is both the inclusion of a second predator and the use of Caputo derivatives. Although LV systems have been extensively investigated, there are no previous studies on our proposed nonlinear modifications. The Grünwald–Letnikov method was applied to find numerical solutions for the proposed model to validate our findings. Several different cases were studied to reach confirmed conclusions. The effect of the variation of fractional-order derivative is handled in this study. From all the results, the effect of adding the small immigration terms is positive for stability. Hence, it can be concluded that the proposed small immigration terms can stabilize the natural populations of the one prey-two predators’ model. An optimal control strategy for the proposed model is shown at the end of the paper. |
| format | Article |
| id | doaj-art-a4183e409b004fa48c2be1cea56986c0 |
| institution | DOAJ |
| issn | 2314-4785 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-a4183e409b004fa48c2be1cea56986c02025-08-20T03:18:59ZengWileyJournal of Mathematics2314-47852023-01-01202310.1155/2023/8795660Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration EffectA. El-Mesady0Omar Bazighifan1Serkan Araci2Department of Physics and Engineering MathematicsDepartment of MathematicsDepartment of Basic SciencesFor over a hundred years, the model of prey–predator has been handled extensively. It has been proved that Lotka–Volterra (LV) models are not stable from all previous studies. In these models, there is divergent extinction of one species or cyclic oscillation. There is no solution for asymptotic stability for the prey–predator models. In this paper, a Caputo fractional-order one prey-two predators’ model with an immigration effect is investigated. Then, the effect of adding a small immigration term to the prey or predators’ populations is studied. The extension is both the inclusion of a second predator and the use of Caputo derivatives. Although LV systems have been extensively investigated, there are no previous studies on our proposed nonlinear modifications. The Grünwald–Letnikov method was applied to find numerical solutions for the proposed model to validate our findings. Several different cases were studied to reach confirmed conclusions. The effect of the variation of fractional-order derivative is handled in this study. From all the results, the effect of adding the small immigration terms is positive for stability. Hence, it can be concluded that the proposed small immigration terms can stabilize the natural populations of the one prey-two predators’ model. An optimal control strategy for the proposed model is shown at the end of the paper.http://dx.doi.org/10.1155/2023/8795660 |
| spellingShingle | A. El-Mesady Omar Bazighifan Serkan Araci Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect Journal of Mathematics |
| title | Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect |
| title_full | Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect |
| title_fullStr | Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect |
| title_full_unstemmed | Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect |
| title_short | Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect |
| title_sort | insight into caputo fractional order extension of lotka volterra model with emphasis on immigration effect |
| url | http://dx.doi.org/10.1155/2023/8795660 |
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